Before exploring the Feedback Vertex Set problem further, we should mention the related Vertex Cover problem.
It asks for a minimum set of vertices hitting all edges in a graph.
While a kernel in the solution size with a linear number of vertices can be obtained using various techniques [1]}, [2]}, [3]}, [4]}, [5]}, a polynomial kernel in a structurally smaller parameter was only discovered in 2011, when Jansen and Bodlaender developed a polynomial kernel in the feedback vertex number of a graph [6]}.
From there, many polynomial kernels for Vertex Cover were discovered in modulators to even larger graph classes [7]}, [8]}, [9]}, [10]}.
In 2020, Bougeret, Jansen and Sau proved the following characterization under common hardness assumptions: Vertex Cover admits a polynomial kernel in the size of a modulator to a minor-closed graph family \(\mathcal {G}\)  if and only if \(\mathcal {G}\)  has bounded bridge-depth [11]}.
With this result, they generalized all existing work on kernels in the size of modulators to minor-closed graph families, and they proved that their results cannot be improved further under common hardness assumptions.
